Parabolic kinetics

Parabolic Reaction Kinetics Parabolic reaction kinetics describe high-temperature corrosion where the rate determining step is diffusion. This is often associated with protective scales and with internal penetration by such corrosive species as oxygen, carbon, or sulfur in certain materials. 

The oxidation of a particular metal in air is limited by the outward diffusion of metallic ions through an unbroken surface film of one species of oxide. Assume that the concentration of metallic ions in the film immediately next to the metal is Cj, and that the concentration of ions in the film immediately next to the air is C2, where and C2 are constants. Use Tick s First Law to show that the oxidation of the metal should satisfy parabolic kinetics, with weight gain Am given by... 

The oxidation of another metal is limited by the outward flow of electrons through a uniform, unbroken oxide film. Assume that the electrical potential in the film immediately next to the metal is Vi, and the potential immediately next to the free surface is Vi, where Vj and Vi are constants. Use Ohm s Law to show that parabolic kinetics should apply in this case also. 

The kinetics of oxidation of mild steel at high temperature are parabolic, with... 

In practice, thermal cycling rather than isothermal conditions more frequently occurs, leading to a deviation from steady state thermodynamic conditions and introducing kinetic modifications. Lattice expansion and contraction, the development of stresses and the production of voids at the alloy-oxide interface, as well as temperature-induced compositional changes, can all give rise to further complications. The resulting loss of scale adhesion and spalling may lead to breakaway oxidation " in which linear oxidation replaces parabolic oxidation (see Section 1.10). 

The aforementioned inconsistencies between the paralinear model and actual observations point to the possibility that there is a different mechanism altogether. The common feature of these metals, and their distinction from cerium, is their facility for dissolving oxygen. The relationship between this process and an oxidation rate which changes from parabolic to a linear value was first established by Wallwork and Jenkins from work on the oxidation of titanium. These authors were able to determine the oxygen distribution in the metal phase by microhardness traverses across metallographic sections comparison of the results with the oxidation kinetics showed that the rate became linear when the metal surface reached oxygen... 

The oxidation of aluminium at room temperature is reported to conform to an inverse logarithmic equation for growth periods up to 5 years duration. At elevated temperatures, oxidation studies over shorter periods illustrate conformity to parabolic, linear and logarithmic relationships according to time and temperature. These kinetic variations are attributed to different mechanisms of film formation . ... 

If the PBR is less than unity, the oxide will be non-protective and oxidation will follow a linear rate law, governed by surface reaction kinetics. However, if the PBR is greater than unity, then a protective oxide scale may form and oxidation will follow a reaction rate law governed by the speed of transport of metal or environmental species through the scale. Then the degree of conversion of metal to oxide will be dependent upon the time for which the reaction is allowed to proceed. For a diffusion-controlled process, integration of Pick s First Law of Diffusion with respect to time yields the classic Tammann relationship commonly referred to as the Parabolic Rate Law ... 

In some circumstances, the reaction rates may not be exactly parabolic, and even initially parabolic rates may be influenced by changes within the oxide scale with time. As an oxide scale grows, the build-up of inherent growth stresses, externally applied strains and chemical changes to either oxide scale or metal may all compromise the initial protection offered by the scale, leading to scale breakdown and ultimately partial or complete loss of protection paralinear, or linear kinetics may ensue. In other circumstances, as will be seen later in this chapter, very small additions of contaminants to... 

A number of authors have reported studies of the oxidation of low-alloy steel in steam. The general observations indicate strong similarities with oxidation in air, the kinetics being typically parabolic and the scales typically comprising FcjO, Fej04, FeO and spinel phases, dependent upon alloy composition, temperature and oxygen partial pressure of the environment... 

Sulphur Trioxide (SO2 -I- O2) Linear reaction rates are observed due to phase boundary control by adsorption of the reactant, SO3. Maximum rates of reaction occur at a SO2/O2 ratio of 2 1 where the SO3 partial pressure is also at a maximum. With increasing 02 S02 ratio the kinetics change from linear to parabolic and ultimately, of course, approach the behaviour of the Ni/NiO system. At constant gas composition and pressure, the reaction also reaches a maximum with increasing temperature due to the decreasing SO3 partial pressure with increasing temperature, so that NiS04 formation is no longer possible and the reaction rate falls. 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

Laqua et al. [1292] have made a detailed kinetic study of the interaction between CoO and 3-Ga203. In this reaction of the second kind (i.e. each reactant has been presaturated with the other), growth of the spinel product layer is parabolic and E = 300 kJ mole-1. 

Kutty and Murthy [1159] have made a kinetic study of the solid—solid reaction between tricalcium phosphate and urea nitrate, a process of possible technological importance. A reduction in particle size, notably of Ca3(P04)2, increased the rate of reaction in powder mixtures and also changed the kinetic characteristics (318—338 K). Reaction in relatively coarse material (between —180 and +200 mesh) obeyed the parabolic... 

However, mathematics is essential to explain how structural data are derived from EXAFS. The EXAFS function, x(k), is extracted from the X-ray absorption spectrum in Fig. 4.10 by removing the approximately parabolic background and the step, i.e. the spectrum of the free atom. As in any scattering experiment, it is customary to express the signal as a function of the wavenumber, k, rather than of energy. The relation between k and the kinetic energy of the photoelectron is ... 

The precise and, where needed, short setting of the residence time allows one to process oxidations at the kinetic limits. The residence time distributions are identical within various parallel micro channels in an array, at least in an ideal case. A further aspect relates to the flow profile within one micro channel. So far, work has only been aimed at the interplay between axial and radial dispersion and its consequences on the flow profile, i.e. changing from parabolic to more plug type. This effect waits to be further exploited. 

Figure 6.26. Kinetics of Mn transformations (described by the parabolic diffusion equation) among three main labile solid-phase fractions of Israeli arid soils. Soils were incubated at the saturated paste regime (after Han and Banin, 1996. Reprinted from Soil Sci Soc Am J, 60, Han F.X., Banin A., Solid-phase manganese fractionation changes in saturated arid-zone soils Pathways and kinetics, p 1079, Copyright (1996), with permission from Soil Sci Soc Am)...

Example 5-2 Kinetic Energy Correction Factor for Laminar Flow of a Newtonian Fluid. We will show later that the velocity profile for the laminar flow of a Newtonian fluid in fully developed flow in a circular tube is parabolic. Because the velocity is zero at the wall of the tube and maximum in the center, the equation for the profile is... 

Evaluate the kinetic energy correction factor a in Bernoulli s equation for turbulent flow assuming that the 1/7 power law velocity profile [Eq. (6-36)] is valid. Repeat this for laminar flow of a Newtonian fluid in a tube, for which the velocity profile is parabolic. 

A comparative analysis of the kinetics of the reactions of atoms and radicals with paraffinic (R1 ), olefinic (R2H), and aromatic alkyl-substituted (R3H) hydrocarbons within the framework of the parabolic model permitted a new important conclusion. It was found that the tt-C—C bond occupying the a-position relative to the attacked C—H bond increases the activation energy for thermally neutral reaction [11]. The corresponding results are presented in Table 6.9. 

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